Finding Factors

Filling in such a table given the headers would be a straightforward task. This problem turns that around and invites students to deduce the headers from the table entries, with the added challenge of trying to reveal as few cells as possible.
 

 

This task works best if each pair of students has access to a computer or tablet so they can work on the interactivity. They may also need pencil and paper so that as they uncover each cell they can work out the factorisation of each quadratic. The first interactivity is a four by four grid where each cell contains a quadratic of the form $x^2+ax+b$ which can be factorised. Give students some time to work on the interactivity. When they think they have correctly identified the column and row headers, they can click "Show the solution" to see if they are right.

Once they are confident at tackling these examples, they could try one of the larger grids mentioned in the problem, where the quadratics are of the form $ax^2+bx+c$.

If individual computers are not available, this could be done as a whole class activity where a few cells are revealed and students are invited to work out as many factors as they can, before requesting the revealing of further cells. Alternatively, students could create their own grids by choosing pairs of brackets for the two columns, multiplying them out, and then revealing certain cells to a partner.
 

What does the coefficient of $x$ tell you about the numbers in the headers?

 

(For the larger grids) What does the coefficient of $x^2$ tell you about the headers?